# MA530_JAN09 - = { x + iy : | x | &amp;amp;lt; / 4 ,-...

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MATH 530 Qualifying exam January 2009 (A. Eremenko) Each problem is worth 10 points. You can use any theorem proved in class or a theorem from the textbook if you state it completely and correctly. 1. Let f n be a sequence of injective analytic functions in the open unit disc, and suppose that f = lim n →∞ f n uniformly on compact subsets of the unit disc. Prove that f is either injective or constant. 2. Let Ω be a bounded region in the plane and f : Ω Ω an analytic function that maps Ω into itself. Suppose that there exists a point z 0 Ω such that f ( z 0 ) = z 0 . Prove that | f 0 ( z 0 ) | ≤ 1. Hint: If a function maps something into itself, iterating it is a good idea . 3. Consider the Taylor expansion in a neighborhood of the point i : z cot z = X n =0 c n ( z - i ) n . What is the radius of convergence of the series in the right hand side? 4. Let f be an analytic function in the strip

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Unformatted text preview: = { x + iy : | x | &lt; / 4 ,- &lt; y &lt; } , and suppose that | f ( z ) | 1 and f (0) = 0. Prove that | f ( z ) | | tan z | for all z . 5. Find a harmonic function in the region { z : | z | &lt; 1 , Im z &gt; } whose boundary values are 1 on the interval (-1 , 1) and 0 on the half-circle. 6. Let f and g be two analytic functions in some region D , and suppose that f ( z ) + g ( z ) is real for all z D . Prove that f-g is constant. 7. For all real a , evaluate Z tan( x + ia ) dx. 1 Use the principal value when a = 0. 8. Let f be a meromorphic function in a neighborhood of 0 and an analytic function in a neighborhood of 0 with the properties (0) = 0 and (0) 6 = 0. Prove that res f = res [( f ) ] . 2...
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MA530_JAN09 - = { x + iy : | x | &amp;amp;lt; / 4 ,-...

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